To unlock Flashpoints in Destiny 2 you must first do Cayde-6's quest line "Patrols" which is unlocked automatically later into the story. Upon completing the Patrols quest you get the Flashpoint quest from Cayde-6. It requires you to beat 6 public events.

After "Main Mission 13: Fury" (near the end of the story) you must talk to Suraya Hawthorne in the farm. This cannot be missed.

Afterwards you can unlock Strikes, Challenges and Patrols. Talk to Cayde-6 in the farm to unlock his patrol quest.
Cayde-6's quest (milestone) is called "Patrols" and takes place in the EDZ. It will be marked on the map by a blue icon. Complete this quest! It's a fairly long quest line and will take around 30 minutes.

After the initial Patrols quest is done, you must complete 3 patrols in the EDZ. There are plenty around the church. You can see them when pressing touchpad (they show up as white icons in the game world).

Return to Cayde-6 to hand in the quest. He gives you a blue engram. This unlocks the "Flashpoint" quest.

Complete the Flashpoint. The objective is to complete 6 Public Events or 4 Heroic Public Events.

Return to Cayde-6 again to complete the Flashpoint and earn a purple engram. This unlocks the trophy.

IMPORTANT: Do not pick up the Liminous Planetary Engram just yet. You should wait with this until you are power level 265 or higher. The reason being that it scales with your power level. Even after completing the quest you can just let it sit with Cayde and the engram's power will increase. These luminous engrams are the only reliable way to get past 265 power.

Side Note: After the story Cayde-6 is found in the Hangar area of the Tower.